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Abstract
We present a more general homological characterization of the direct summand theorem (DST). Specifically, we state two new conjectures: the socle-parameter conjecture (SPC) in its weak and strong forms. We give a proof for the weak form by showing that it is equivalent to the DST. Furthermore, we prove the SPC in its strong form for the case when the multiplicity of the parameters is smaller than or equal to two. Finally, we present a new proof of the DST in the equicharacteristic case, based on the techniques thus developed.
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